ON A SPECIAL CLASS OF p.GROUPS
BY
N. B L A C K B U R N Manchester, England
I n t h e s t u d y of p-groups t h e chief difficulty lies in t h e fact t h a t t h e n u m b e r of such groups is v e r y large. I t is therefore of interest to s t u d y certain classes of p-groups, a n d the present paper is d e v o t e d to such a topic.
L e t G be a g r o u p a n d let x, y be elements of G. W e define t h e c o m m u t a t o r [x, y] a n d t h e t r a n s f o r m xz b y t h e formulae:
[x, y] = x - l y - l x y , x ~ = x[x, y] = y - l x y .
F o r subsets U, V of G, [U, V] denotes t h e g r o u p generated b y all c o m m u t a t o r s [u, v], where u E U, v E V. W e define t h e lowe~: central series
G >1 ?2(G) >i ?s(G) >~... >1 ?~-1 (G) >1 ?f(G) >7...
o f G i n d u c t i v e l y as follows:
? 2 ( G ) = [ G , G ] , ?,(G) = [?,_I (G), G] (i = 3 , 4 . . . . ).
I f there exists an integer k such t h a t ?k (G) = l, t h e n G is said to be nilpotent, a n d if k is the smallest such integer, k - 1 is called t h e c/ass of G.
p is t o d e n o t e a prime n u m b e r a n d a p - g r o u p is a group of order a power of p. I t is well k n o w n t h a t all p-groups are nilpotent, a n d we m a y therefore speak of t h e class of a p-group. I f m, n are integers a n d 3 ~< m ~< n, it is convenient t o denote b y CF (m, n, p) t h e set of all groups G of order p" a n d class m - 1 in which
(~,-1 (G) : ~, (G)) = p (i = 3, 4 . . . . m).
Similarly E C F (m, n, p) denotes the set of those groups G of CF (m, n, p) in which G / ? 2 (G) is e l e m e n t a r y Abelian. These t w o classes of groups are to be investigated. M a n y of o u r earlier results can also be s t a t e d for a n o t h e r class of groups which we d e n o t e b y N C F (m), a n d which consists of all nilpotent groups G of class m - 1 in which each of t h e groups
?I_I(G)/?~(G) (i = 3, 4 . . . m) is an infinite cyclic group. The general considerations on
4 6 N. BLACKBURN
which our investigation of these groups is founded are based on a paper of P. Hall [3]
together with a few remarks which are developed in w 1.
Perhaps t h e most interesting groups considered are the p-groups of maximal class, t h a t is, the p-groups of the greatest class which is compatible with their order. I n w 2 we begin b y discussing the most elementary properties of the~e groups and their generaliza- tions to the classes of groups considered. Two characteristic subgroups are then intro- duced and these play a fundamental p a r t in the discussion which follows. Our aim is to find what we call the degree of e o m m u t a t i v i t y of our groups and, in particular, to find whether or not this is greater t h a n 0. The main result is Theorem 2.11 which asserts t h a t a considerable proportion of the groups in question h a v e degree of c o m m u t a t i v i t y greater t h a n 0. We conclude w 2 b y a result (Theorem 2.16) on the m a x i m a l n u m b e r of generators of the derived group of a group of
CF(m,n,p).
I n w 3 we consider the groups of
ECF(m,n,p)
a n d show t h a t their s t u d y reduces essentially to t h a t of p-groups of m a x i m a l class. The problem of finding the degree of c o m m u t a t i v i t y is continued and results in this direction are Theorems 3.8, 3.10, 3.12, 3.13 and 3.14. The power-structure of these groups is also investigated. I n w 4 all groups of order p~ a n d class 5 and all 3-groups of m a x i m a l class are found.A p a r t from their intrinsic interest p-groups of maximal class are also of interest on account of certain applications. Thus it is sometimes the case t h a t in characterizing all p-groups with a given simple property, we find t h a t these groups generally have a simple structure but t h a t exceptionally certain p-groups of m a x i m a l class also have the given property. The best-known instance of this is the problem of finding all p-groups with just 1 subgroup of o r d e r p: such a group is either cyclic or is a generalized quaternion group (which is a 2-group of m a x i m a l class). The consideration of such problems tends to take us outside the scope of the present work and is accordingly not discussed here. The author hopes, however, to return to this question in a later paper.
A paper on p-groups of maximal class has already appeared, namely b y A. W i m a n [12].
The discussion given in the present work is to some extent based on W i m a n ' s ideas, and we wish to express our indebtedness to this author. Unfortunately the conclusions t h a t we have reached do not coincide with those of Wiman, and so we h a v e made the present work independent of [12]. A detailed comparison will not be made, b u t it m a y be said t h a t it is in I I a n d I I I of [12] t h a t statements are made which seem to us to be in general untrue.
The author also wishes to express his deep gratitude to Prof. P. Hall, of King's College, Cambridge, whose suggestions and encouragement were of the greatest value to him when he was working on the material of this work.
oN A SPECIAL CLASS OF p-OROUPS 47 t . W e begin b y s t a t i n g s o m e of t h e k n o w n results which are of f u n d a m e n t a l i m p o r - t a n c e for o u r purpose. A m o n g s t these t h e following t h e o r e m p l a y s a n i m p o r t a n t p a r t in t h e construction of n i l p o t e n t groups.
T H E O R E M 1.1. Let G be a group 9enerated by a set X o[ elements. I] Y @ a set o/ele- ments which together with ~, ~1 ( G) generate 7, ( O), then ~, ~t ((7) is generated by Y,+2 ( G) together urith all commutators Ix, y], where x , y run through X , Y respectively. This is true [or i = 1,
2 . . . provided that yl(G) 48 interpreted to mean G.
~ o r t h e proof we refer t h e r e a d e r t o P. H a l l [3], T h e o r e m 2.81.
T h e following result is due t o L. K a l o u j n i n e [7] a n d [8].
T H E O R E M 1.2. Let G be a group, and let
H = H o
~>H,
~ > H 2 >7... ~H, >~...be a series o/ normal subgroups o] G. I[ L is any subgroup o[ O such that [L, H,_I] ~< H i (i = 1, 2 .... ), then
[9b(L), H,] ~< H,+j (i = 0, 1 .... ; j = 2, 3 .... ).
W e shall need a n application of this involving a n o t h e r characteristic subgroup. F o r a n y g r o u p (7, we define 7, ((7) to be t h a t s u b g r o u p of (7 for which ,/, (G)/y, ((7) is t h e centre of G/},,((7) (i = 2, 3 .... ). T h u s ~2(G) = (7, a n d for i > 2 ~/,((7)/> y,_l(G).
I f in T h e o r e m 1.2 we p u t L = (7, H = ~, ((7), a n d define t h e s u b g r o u p s H~ i n d u c t i v e l y b y the rules H 0 = H, a n d Hk~ , = [H~, G] for k>~ 0, we find t h a t [),j(G), Hk] ~< Hs+k. B u t b y t h e definition of ,1,, H 1 ~ 7,(G); hence, Hj <<.7,+j_, ((7), for j >/1. Thus,
[yj(G), 7,((7)] = [~J((7), H0] < Hj < Y,+s-, (G).
C O R O L L A R Y 1. I n any group G, [Tj(G), ~?,(G)] <<. },t+r (G) (i, j = 2 , 3 .... ).
Using },, (G) ~< y,+,(G) we o b t a i n t h e well k n o w n result:
C O R O L L A R Y 2. I n any group (7, @,((7), rj(G)] < r l + j ( G ) (i, j = 2 , 3, ...).
W e shall also need t h e upper central series,
1 = ~'o (G) < $, (G) ~< ... ~< ~|-1 (G) < ~1 ((7) ~< . . . .
of a group G, which is defined i n d u c t i v e l y b y t h e rules: $ 0 ( G ) = 1, ~,(G)/$~_I(G) is the centre of G/~,_ 1 (G) (4 = 1, 2 .... ). T h e n (7 is u l l p o t e n t of class m - 1 if a n d only if ~m-1 ((7)
= (7, $~_~ (G) # (7, a n d in such a group 7, (G) ~< ~m-, (G), b u t y~ (G) $ $~-,-1 (G). I t follows t h a t for i = 1, 2 . . . . , m - l,
[G, ~,+~(G)] < 7,+, (G) < ~m-,-~ ((7), a n d so ~Y,+t (G) ~< ~m-, (G).
4 8 1~. B L A C K B U E N
I n T h e o r e m 1.2 we m a y p u t L = G, H i = ~k_z(G) (l = 0,1 .... , k) a n d H i = 1 (l >/~).
W e find t h a t for 0 ~< ~ + l < k,
[7~(G), ~_l(G)] <
~_,_l(G),
or, p u t t i n g i =/c - 1:
C O r O L L a r Y 3. I n a n y group G, [7~(G), $~(G)] ~< $~_~(G) (2 ~< i ~< i).
F u r t h e r , for a n y g r o u p G we d e n o t e b y
G >~ G' >~ G" ~ ... >~ G (t-1) >1 G ~f~ >1 ...
t h e derived series of G, which is defined inductively b y
G ' = [ G , G ] = y z ( G ) , G ( 0 = [ G (t-1), G (f-l)] ( i = 2 , 3 . . . . );
also we denote b y (I)(G) t h e F r a t t i n i s u b g r o u p of G.
I f xl, xa . . . x~ are elements of G we define the simple commutator Ix1, xz . . . xn] b y induction on n; for n = 2 it is alreaxiy defined, a n d for n > 2 we p u t
[Xl, X2, . . . , Xn] : [[Xl, X2 . . . . , Xn-1], Xn].
I n t h e sequel we shall o n l y be concerned w i t h one of the s u b g r o u p s ~/t (G), n a m e l y
~3{G), a n d we shall therefore p u t ~ ( G ) = ~a(G). This s u b g r o u p possesses t h e following p r o p e r t y .
T H ~ o ~ w M 1.3. Suppose that G is a nilpotent group and that H is a subgroup o / G with the property that H ~ (G) = G. Then H is normal in G and ~t (H) = ~t ( G ) , / o r i = 2, 3 . . .
I t is obvious t h a t 71 (H) ~< 7t (G). L e t m - 1 be t h e class of G: we p r o v e t h a t 7t (H) = 7t(G) b y induction on m - i. F o r i = m it is trivial, since 7t(H), ~i(G) are t h e n b o t h equal to the u n i t subgroup. F o r i < m it follows f r o m t h e i n d u c t i v e h y p o t h e s i s t h a t 7~+1(H) = T i l l ( G ) . N o w b y hypothesis, G is g e n e r a t e d b y t h e elements of ~(G) a n d H ; hence b y T h e o r e m 1.1, 7~(G) is g e n e r a t e d b y ~t f l (G) a n d all c o m m u t a t o r s
Y = [ Y l , Y~ .. . . . Yi],
where each c o m p o n e n t Yt of y is either a n e l e m e n t of ~ (G) or of H . B u t b y T h e o r e m 1.2 Corollary I, [~/(G), ~j(G)] <~j+2(G) (~ = 2, 3 . . . m - 2 ) , a n d so a n y c o m m u t a t o r y, one of whose c o m p o n e n t s is a n e l e m e n t of ~ (G), is a n e l e m e n t of ~t+1 (G). H e n c e ~l (G) is g e n e r a t e d b y 7 t + 1 ( G ) = ~ + I ( H ) a n d all c o m m u t a t o r s of t h e f o r m of y, where y j E H . Since each of these c o m m u t a t o r s belongs to ~'l (H), it follows t h a t 7t (G) ~< 7~ (H), a n d therefore yt (G) = yt (H), as required. I n p a r t i c u l a r H ~> 72 (H) = Y2 (G), a n d so H is a n o r m a l s u b g r o u p of G.
I n the m a n i p u l a t i o n of c o m m u t a t o r s t h e following f o r m u l a e are largely used:
[xy, z] = Ix, zJ~[y, z], Ix, yz] = [x, z] Ix, y]~, (1)
O N A S P E C I A L C L A S S O F ~ - G R O U P S 49 where x, y, z are any elements of any group. We also note t h a t
[x -1, y] = [y, x] ~-', [x, y-i] = [y, x]~-'. (2) By (1), it follows t h a t if one of the components x or y is multiplied by an element of the centre of the group in which t h e y lie, then the value of [x, y] remains unaltered.
We shall also require results on the powers of a product of elements, of which the simplest is the following.
THEORZM 1.4. I / G is a group, x E G and yETr(G), t h e n / o r a n y integer n,
[x",y]--[x, yn]~[x,y]" (rood r,+~(G)),
(xy)=----x"y"[x, y]-(~) (mod 7r+a(G)).
T h i s is t r u e / o r r = 1, 2 . . . i/ 71 (G) is interpreted to mean G.
This is proved by a simple induction on n, using (1) and (2).
We shall require the following consequences of Theorem 1.4.
T H E O R E ~ 1.5. Let G be a p-group.
(i) I / n l , n 2 .. . . . n~ are the invariants o / t h e Abelian p-group G/Ta(G), and n I ~ ng.
9 .. >1 n,, then the exponent o/72 (G)/73 (G) i8 aS most p,2..
(if) / ] / o r some i >~ 2 7~(G)/7~+ l (G ) is o/ exponent p", then the exponent o/ 7~+ l (G) /Tt§ 2 (G) is at most 19".
( iii ) I / G / 7 z ( G) is an A belian group with two invariants 2,/~, where ~ >1/~, then 72 ( G) / 7a ( G ) is a cyclic group o / o r d e r aS most p~.
To prove (i), suppose t h a t G/72 (G) is the direct product of the cyclic groups generated by the elements x~72(G ) (i = 1, 2 ... r), where pn~ is the order of x~ modulo 72(G). Then G is generated by Xl, x 2 ... x~ together with the elements of 72 (G), and it is easily deduced from Theorem 1.1 t h a t 72(G) is generated by 73(G) and all the commutators [x~, xj]
In such a commutator j>~2 and so n s ~ n 2 ; hence x[~'ET2(G), and by (1 < i < ] < r ) .
Theorem 1.4
[x~, xj]Pn'----[x~, x ~ n ' ] ~ l (rood 73(G)).
72(G)/Ta(G) is therefore of exponent at most p'~.
The proof of (if) is very similar: if G is generated by Yl, yg. ... Ys and 7~ (G) is generated by Zl, Z 2 .. . . . z t and 7~+l(G), so t h a t by hypothesis zF"ET~+I(G), then by Theorem 1.1 7~+1(G) is generated by all commutators [y, zs] and 7~+9.(G), whilst by Theorem 1.4,
pm 7) m
[y~,zj] ----[y, zs ]~-1 (modT~+2(G)).
5 0 N. B L A C K B U R N
To prove (iii), we observe as in (i) t h a t if x, y and
?2(G)
generate G, then [x, y] a n d ?a(G) generate 72 (G), so t h a t ?2 (G)/?3 (G) is cyclic. The bound on the order of this group follows from (i).We conclude the present p a r a g r a p h b y a r e m a r k on a formula oi P. Hall for a power of a product (see [3], w 3). I f x, y are elements of a group, we define elements ~ (x, y) of this group inductively b y the rules.
(~l(x, y) = y, a~(x, y) = [at_l(x, y), x] (i = 2, 3 .... ).
Also, if G is a p-group, we denote b y P,(G) (i = 0 , 1 .... ) the subgroup of G generated b y all pi-th powers of elements of G. I n this notation we have the following result.
T H E O R E M 1.6. Let G be a p-group, and let x, y be elements o/ G. Denote by Y the group generated by y and ?~ (G). T h e n
(D if)
(xy)J'=--x ~ (~ (~2 ...(~t ...G~
p - 2
(mod P1
(Y') [Y, ?p-1 (G)] ]-I [?, (G), ?p_, (G)]),where a~ = a~(x, y). (For p = 2 we interpret ?~_I(G) to m e a n
?2(G)).
F o r p = 2 this is trivial, since
(xy)2 = x20-120-2 [G2, (71],
and so we need only consider the case when p is odd. The proof in this case is a slight modification of the proof given b y Hall for his formula. According to this proof, it is necessary first of all to arrange the various distinct c o m m u t a t o r s of x a n d y in an order.
L e t C~l, cw2 .. . . . c~q~ be the c o m m u t a t o r s of x a n d y of weight w, other t h a n ~w. The order t h a t we t a k e is
0"2~ G3~ C31~ 9 9 9 C3q,~ 9 9 9 ~ G w ~ V w l ~ C w 2 , 9 9 9 ~ C w q w , . . . .
I f G is of class m - 1, all commutators of weight m are equal to 1, a n d we only consider those aw, cwj for which w < m.
The proof proceeds b y stages, each stage being attained b y a n u m b e r of steps. At the 0th stage, we have
(xy) p = x a l x a l . . , xal,
and the only point a t which we wish to refine the procedure of Hall is in passing to the 1st stage. This is done b y collecting all the elements x in the expression XalXal ... x a l over on the left. To do this we m a y begin with the following steps:
o ~ x SPECIAL CLASS OF p-GROVTS 51
( X 0 " I ) p = X 2 0 " l [ a l , X J O ' l X O ' I . . . X O " 1
= x ~ a l [al,X]Xa 1 [ql,X] (11 ... x a 1
= X 2 a l x [ a x , x ] [ a l , x , x ] a l [al,X](~l . . . x a l
= x3 ~1 [~1, x]~ [al, x, x] al [a~, x] a ~ . . . x a~
H e r e we h a v e collected t h e first three e l e m e n t s x a n d o b t a i n e d (xy) p = x3 al a~2a3al a2al . . . x a l .
S u p p o s e t h a t after collecting t h e first i e l e m e n t s x, where 1 < i < p, we h a v e a n expression of t h e f o r m
( x y ) ~ = x t d l d 2 . . . d~ x a l x a l . . . X a l , (3) i n which n~(i) of t h e dl are e q u a l to a~, for w = 1, 2 . . . m - 1. T h e n
n~(1) = 1 n w ( 1 ) = O ( w = 2 , 3 , . . . , m - 1). (4) I f i < p we c a n collect a n o t h e r x over o n t h e left, a n d o b t a i n f r o m (3)
( x y ) ~ = xl+ l dl [dl, x] d~[d 2, x] . . . dk [dk, x ] a l x a l . . . x a l .
Therefore n I (i + 1) = n I (i) + 1 (i = 1, 2 . . . p - 1), (5)
n w ( i + l ) = n w ( i ) + n w _ l ( i ) ( i = 1 , 2 . . . p - l ; w = 2 , 3 . . . m - l ) . (6) B y i n d u c t i o n o n i, (4), (5) a n d (6) show t h a t
n w ( i ) = ( i w ) ( i = l , 2 , . . . , p ; w = l , 2 . . . m - l ) .
T h u s a t t h e e n d of t h e first stage, t h a t is, w h e n all t h e p e l e m e n t s x h a v e b e e n collected o n t h e left, we h a v e a n e q u a t i o n of t h e form
( x y ) ~ = x~ el e2 .. . el, (7)
w h e r e ( P w ) o f t h e e , a r e e q u a l t o a ~ . I n p a r t i c u l a r , j u s t o n e o f t h e e j i s e q u a l t o ~ .
T h e n e x t stage is t o collect t h e e l e m e n t s a~ which occur i n (7), t h e n t h e e l e m e n t s a~, a n d so on. A t e~eh step i n each stage a n e w c o m m u t a t o r is i n t r o d u c e d , which c a n be w r i t t e n as a c o m m u t a t o r i n a 1, a~, . . . , ap. All c o m m u t a t o r s of weight less t h a n m are collected after a finite n u m b e r of stages, a n d this is t h e e n d of t h e process. T h e expo- n e n t w i t h which each of t h e c o m m u t a t o r s a p p e a r s is c a l c u l a t e d b y Hall, a n d u s i n g his
5 2 N . B L A C K B U R N
r e s u l t , we f i n d t h a t
(xy) =x" al az as ~31 . . . u S a , . . . O w (~Wl V w 2 . . . C w q w . . . r (8)
where, for 3 ~< w < p, nwj is d i v i s i b l e b y p.
N o w since e a c h c o m m u t a t o r c~j w h i c h occurs w i t h p o s i t i v e e x p o n e n t in (8) can be w r i t t e n a s a c o m m u t a t o r of w e i g h t g r e a t e r t h a n 1 i n al, as, . . . , a t , i t follows t h a t such a c~j lies in Y', a n d so, for 3 ~< w < p,
c:~JeP 1 (Y').
F o r w ~> p, we can write c~j = [a, b] s a y , w h e r e a a n d b a r e also c o m m u t a t o r s in aL, a2 . . . . a~.
I f a is of w e i g h t i i n x a n d y, t h e n b is of w e i g h t a t l e a s t p - i in x a n d y. T h u s if 2 <~ i ~<
p - 2 , a 6 ~ ( G ) , b67r_~(G), a n d
c~j e [7, (G), r~-, (G)].
I f i = 1, t h e n . a is e i t h e r x or y, a n d since i t c a n be e x p r e s s e d as a c o m m u t a t o r in al, a2, . . . . a~, i t follows t h a t a = y = al, a n d so a 6 Y. Also b ETp_ 1 (G), since p is o d d , a n d
cwj e [ Y, ~_~ (G)].
I f i > / p - 1, t h e n a 6 ~ r _ l ( G ) , a n d b is a c o m m u t a t o r in al, a s , - - - , a~, a n d t h e r e f o r e lies in Y. T h u s we a g a i n o b t a i n
cwj e [ Y, r~-~ (G)].
Since P I ( Y ' ) , [Y, 7~_1(G)] a n d all [ ~ ( G ) , ~_~(G)] (i = 2, 3 . . . p - 2) a r e n o r m a l sub- g r o u p s of G, t h e t h e o r e m follows f r o m (8).
2. W e b e g i n b y f i n d i n g t h e m a x i m a l class of a p - g r o u p of g i v e n order. F o r t h i s we n e e d t h e following r e m a r k .
L ~ M M A 2.1. I / G is a group and N is a normal subgroup o / G / o r which G I N is cyclic, then G' = [G, •].
I t is o b v i o u s t h a t [G, N ] < G', a n d so in o r d e r to p r o v e t h e l e n ~ n a we m a y a s s u m e t h a t [(7, N ] = 1, t h a t is, t h a t N is c o n t a i n e d in t h e c e n t r e of G. B u t t h e n i t follows t h a t G is A b e l l a n , since G I N is cyclic (see [13], K a p . I V , p. 104), a n d so G' = 1 as r e q u i r e d .
N o w in a n o n - A b e l i a n n i l p o t e n t g r o u p G, y~(G) = G' a n d ys{G) c a n n o t coincide, a n d so b y t a k i n g N = 7~(G) in L e m m a 2.1, we see t h a t G / ~ ( G ) c a n n o t b e cyclic. I n p a r t i c u l a r , in a n o n - A b e l i a n p - g r o u p G, y~ (G) is of i n d e x g r e a t e r t h a n p, a n d if G/$~ (G) is of o r d e r p~, t h e n i t is e l e m e n t a r y A b e l i a n . Also, if G is a T - g r o u p of class m - 1 where m / > 3, t h e n e a c h of t h e g r o u p s ~ - 1 (G)/~t (G) is of o r d e r a t l e a s t p (i = 3, 4, . . . , m), a n d so G is of o r d e r a t l e a s t pro. T h u s a g r o u p of o r d e r pm is of class a t m o s t m - 1.
ON A SPECIAL CLASS OF p-GROUPS 53 W e shall refer to groups of order pm a n d class m - 1 for some m ~> 3 as p-groups o/
m a x i m a l class. I f G is such a group,
(G : ~ ( G ) ) = p~, ()J,_~(a) : ~,(a)) = p (i = 3, 4, . . . , m).
T h u s G has just p + 1 m a x i m a l subgroups, a n d these are all n o r m a l in G. The remaining n o r m a l subgroups are d e t e r m i n e d in t h e simple result:
L z: M M A 2.2. I f G is a p-group of m a x i m a l class and N is a normal subgroup o/ G o/
index pr where r >~ 2, then N = YT (G).
T h e group G I N is of order pT, a n d therefore of class a t m o s t r - 1 ; t h a t is, ~, (G/N) = 1.
B u t
rJ(r = r , ( G ) Y / N (i = 2, 3, ...), a n d so y~ (G) < N. B u t
(G : r r ( a ) ) ~= p ' = ( a : ~v), a n d so y, (G) a n d N , being b o t h of the same order, are equal.
As s t a t e d in t h e i n t r o d u c t i o n we shall consider more general classes of groups. F o r CF(m, n, p) we m a y generalize L e m m a 2.2 as follows:
L E ~ M A 2.3. I / G E C F ( m , n, p) and N is a normal subgroup of G of order pt which is contained in Y2 (G), then N = ym-t (G).
Obviously, yj(G) is of order pm-J. T h u s if i = m - 2, t h e result is obvious, a n d for m - i > 2 we m a y use induction on m - i. Since N < y2(G), there exists a normal sub- group M of G of order p~+l, such t h a t
N < M < 72 (G),
(see [13], K a p . IV, p. 104), a n d b y t h e inductive hypothesis, M =Tm_t_l(G). B u t M / N is a n o r m a l subgroup of G I N of order p, a n d is therefore contained in t h e centre of G / N , t h a t is, [G, M ] < N . H e n c e
N > / [ a , ~,m_,_ ~ ( a ) ] = r ~ _ , ( a ) . B u t N a n d ym-~ (G) are b o t h of order T ~, a n d are therefore equal.
L e m m a 2.2 shows t h a t in a p - g r o u p of m a x i m a l class t h e terms of t h e u p p e r central series a r e t h e same as those of t h e lower central series, b u t in t h e reverse order. W e n o w state t h e corresponding result for t h e groups of CF (m, n, p) a n d N C F (m).
T H E O I ~ E M 2.4. I / G E C F ( m , n, p) or G E N C F ( m ) , then
~-~(G) f~ ~,~(G) =~m_,(O) (i = 0 , 1 . . . m - 2 ) .
5 4 N. BI.,ACKBUR~
I n the case when G E C F ( m , n, p), this is m o s t easily p r o v e d as follows. B y a r e m a r k m a d e in w 1, ~l(G)~>Tm-~(G), b u t ~ t ( G ) ~ ~'m_~_1(G). H e n c e ~ ( G ) N F2(G)>~Fm_~(G), since i -~ m - 2. B u t ff ~ (G) (] 79- (G} were of g r e a t e r order t h a n 7,,_~ (G), it would follow f r o m L e m m a 2.3 t h a t ~t (G) ~ 7~ (G) contains 7~-~-1 (G), a n d this is n o t possible.
I f G E N C F ( m ) , we m u s t a d o p t a different procedure, for L e m m a 2.3 has no dh'ect analogue for these g r o u p s . . I n rids case we proceed b y induction on i: for i -- 0 it is trivial.
F o r i > 0, we observe first t h a t ~(G) fl 72(G) >~Tms~(G), just as a b o v e . N o w suppose t h a t t h e t h e o r e m is n o t true: t h e n t h e r e exists a n e l e m e n t a of $~ (G) N Y2 (G), which does n o t belong t o 7~_~(G). I f / < m - 3, t h e r e exists a n integer r, such t h a t a is a n e l e m e n t of y~(G), bu~ no~ of 7~§ where 2 ~< r < m - i - 1. Since 7~(G)/y~+I(G) is cyclic, we m a y choose a n e l e m e n t x which t o g e t h e r with 7r+l(G) generates y~(G), a n d since a does n o t lie in 7~ ~1 (G), t h e r e exists a non-zero integer, g, such t h a t a = x~y, where y E~,+I (G). N o w let z be a n y eIement of G. B y T h e o r e m 1.4, using (1),
[z, a] = [z, x~ y] ~= [z, y] [z, x~]~ ~ [z, x] ~ (rood 7~+~(G)).
B u t a e $t (G), a n d so [z, a] E $t-1 (G) N ~'2 (G). B y t h e inductive h y p o t h e s i s ~t-1 (G) ~) 72 (G) = 7,,_t~(G). Since m - i + 1 >~ r + 2, it follows t h a t [z, a ] ~ 7, ~z(G), a n d so
[z, x] ~ ~ 1 (rood 7~+2(G)).
B u t b y T h e o r e m 1.1, 7,~ ~ (G) is g e n e r a t e d b y ~,, ~ (G) a n d all elements [z, x] as z runs t h r o u g h G. H e n c e 7~+~(G)/Tr+~(G) is a g r o u p of e x p o n e n t a t m o s t :r Since r + l < m - l , this contradicts t h e definition of N C F ( m ) , a n d so t h e t h e o r e m is p r o v e d for i < m - 3. F o r i = m - 2 , it is, of course, trivial.
F o r the groups u n d e r consideration, t h e case m - 3 is n o t w i t h o u t interest: for e x a m p l e , 0. Sehreier [9] d e t e r m i n e d all t y p e s of groups in E C F (3, n, p). H o w e v e r the considerations of the p r e s e n t work do not a p p l y to this case, a n d we shall henceforth a s s u m e t h a t m > 3.
I n this case t h e basic step is the i n t r o d u c t i o n of a n o t h e r characteristic s u b g r o u p which we shall d e n o t e b y y~(G) (cf. W i m a n [1217. This is defined for a g r o u p G of C F ( m , n, p) or N C F ( m ) , where m > 3, b y t h e p r o p e r t y t h a t 7~(G)/7~(G) is t h e eentraliser in G/~a (G) of 7~ (G)/7*(G); t h a t is, 7~ ((7) is t h e largest s u b g r o u p of G such t h a t
[~,~(a), r~(a)] < r,(a).
T h u s it is clear t h a t 7~ (G) is a characteristic p r o p e r s u b g r o u p of G which c o n t a i n s 7, (G).
I n order to see h o w ~1 (G) is situated in G, we p r o v e t h e following.
L E M M ~ 2.5. Let G be a nilpotent group o / c l a s s m - 1, where m > 3, and 8ulrtx~e that [or i ~- 3, 4 . . . m, 7~-1 (G)/y~ (G) is cyclic. For i = 3, 4 . . . m - 1, let Ks be the subgroup o/
ON A SPECIAL CLASS OF ~-GROUPS 5 5
G delined by the/act that K,/~,,.I(G ) / 8 the central~set in G/~,.I(G) o/r,_~(G)/r,.~(G). Then G/K~ is a cyclic group o/the same order as ~'t (G)/~I+I(G).
To prove this suppose t h a t a is an element which together with ~ ( G ) generates
~t_l(G), a n d t h a t b is an element which together with ~+~(G) generates ~(G). If x is a n y element of G, then [a, x] lies in ~(G), a n d so there exists an integer ~ such t h a t
[a, x ] - - b e (mod r~+l(G)),
or a ~ - - a b ~ (rood ~ 41 (G)).
I f n is the order of ~(G)/~+I(G), and we m a p z into the residue class of integers modulo n containing ~, we obtain a homomorphism of G into the additive group of residue classes modulo n. K~ is the kernel of this homomorphism. The image is the whole group of residue classes, for as x runs through G, b y Theorem 1.1 the elements [a, x] and 7~.l(G) generate
~t(G), and so the elements b ~ together with ~+I(G) generate ~(G). Hence G/K~ is cyclic of order n.
I t follows from L e m m a 2.5 t h a t if G E C F ( m , n, p) (m > 3), then ~I(G) is of index p, whilst if G ENCF(m) (m > 3), then G/yl(G ) is an infinite cyclic group.
T H E O R ~ . ~ 2.6. If GECF(m,n,p) or G E N C F ( m ) ( m > 3 ) , then the derived group
~'1 (G) of ~1 (G) is contained in ~a (G).
To prove this we m a y argue modulo ~4 (G), and m a y therefore assume t h a t m = 4.
~ (G) is generated b y all elements [u, v], as u, v run through ~I(G), and so we have to prove t h a t [u, v] e y3(G). B y the above remarks, there exists an element s of G which together with ~i (G) generates G. I f [u, s] = x, [v, s] = y, then
[u, v]" = [u', v'] = [ux, vy].
B u t x, y are elements of ~2 (G), and it follows from the definition of Yl (G) t h a t t h e y com- m u t e with u, v. Since also ),2(G) is Abelian, we find, using (1), t h a t
[ux, vy] = [u, v].
Thus [u, v] commutes with s. B u t again [u, v], being an element of ~2 (G), commutes with each element of ~'I(G), and so [u, v] lies in the centre of G. I t follows from Theorem 2.4 t h a t [u, v] ~ ~3(G), as required.
N e x t we discuss the position of ~ (G) = ~ (G), as defined in w 1.
T H E O ~ 2.7. 1] G E C F ( m , n, p) (m > 3), then ~(G) /8 a subgroup of ~I(G) o] index p. I / G e N C F ( m ) (m > 3), then ~(G) is a subgroup o/ ~(G), and ~(G)/~(G) is an in/inite cyclic group.
5 6 ~ . BLACKBURN
In both cases we prove t h a t ~1 (G)/~I (G) is cyclic as follows. H s is as defined in the proof of Theorem 2.6, we deduce from Theorem 1.1 t h a t ~2(G) is generated b y ~s(G) to.
gether with the elements [u, s] and [u, v], as u, v run through ~1 (G). B y Theorem 2.6 the elements [u, v] lie in ~s(G), and m a y therefore be disregarded. Let a be an element which together with ~s(G) generates ~2(G). Then there exists a finite set of elements ul, u 2 .. . . .
U r Of ~1 (G), such t h a t
a-:-- I~ [uf, s] at (mod rs(G)),
~-1
where ~tl, A2, -.-, ~ are suitable integers. Let
8 1 =
so t h a t if [81,8 ] = 8~, then by (1) and Theorem 1.4, s~ ~ a (rood Ya (G)). Thus Ye (G) is gen- erated by ~s (G) and s 2.
For any element u of ~I(G), we m a y therefore write [u, 8] --8[; we then define ~ = u 8 ~ ~ Thus ~1 (G) is generated b y 81 and the elements ~. Now ~ E ~1 (G), and so by Theorem 2.6 commutes modulo ~s(G) with each element of
~'I(G).
But also ~ commutes modulo~s(G) with 8, for
[a, s] = [us~ ~, s] = [u, 8] [s; ~, s] ~s.~ -~ "--- 1 (mod ys(G)),
and so ~ lies in the centre of G modulo ~a(G), t h a t is, a E ~ (G). B y Theorem 1.2, Corollary 1, [~(G), ~ ( G ) ] ~<~4(G), and so ~/(G) ~<~I(G). Hence yl(G) is generated b y 81 and ~(G), and ~1 (G)/~I (G) is cyclic.
To find the order of ~I(G)/~ (G), we use the fact that according to Theorem 1.4, for any integer r
[sl, s]:--s~ (rood rs(O)).
B y Theorem 2.6 s~ commutes modulo ~s (G) with each element of ~1 (G), and so 8~ e ~/(G) if and only if 8~ and 8 commute modulo ~s(G), t h a t is, if 8[e~,a(G). If G eCF(m, n, p), it follows that s[ ~r](G), 81 {~ ~ (G), and therefore :~ (G)/~ (G) is of order p. If G ~NCF(m), it follows t h a t no positive power of 81 lies in ~ (G), and so ~I(G)/~ (G) is infinite.
COROLLARY. I / G e C F ( m , n, p) or G e N C F ( m ) , then ~7(G) -- ~m_~(G).
B y a remark in w 1, ~ (G) ~< tm-~ (G). Also, by Theorem 1.2, Corollary 3 and Theorem 2.4, for m > 3,
< N = r e ( G )
and so ~m_~(G)~ ~ ( G ) . Suppose t h a t ~ _ ~ ( G ) > ~/(G) ; then ~_~(G) is of finite index r in
~ (G). Hence 8~ ~ ~_~ (G), and by Theorem 2.4
[8~, 8] E ~m_a(G) N ~2(G) = ~a(G).
ON A SPECIAL CLASS OF p-GROUPS 57
B u t b y Theorem 1.4 [s~, s] =sg. (rood ~a(G)), a n d so s26 ~a(G ). I f G 6 NCF(m) this is im- _ r possible since r is finite, and if G E CF (m, n, p) this implies t h a t r >/p. I n b o t h cases, we h a v e a contradiction, a n d so ~m-~ (G) = ~(G). F o r m = 3 the corollary is trivial.
I n future, instead of ~ (G) we shall speak of the more familiar group ~m-~ (G). We are now therefore considering groups with the following series of characteristic subgroups.
G >~1 (G) > ~_2(G) ~> ~ ( G ) >~3(G) > ... > r~_l(G) >rm(G) = 1.
I n this series all factor groups of successive terms are cyclic of equal order, except for
~m-2(G)/~2 (G). This group is arbitrary, as is seen b y considering the direct product of a group in which it is the unit subgroup with an arbitrary Abelian group.
N o w according to Theorem 1.2, Corollary 2, in a n y group G, [~,(G), ~j(a)] <r,+j(G) (i, i = 2, 3 . . . . ).
This is in general the best possible result, as, for example, the Sylow p-subgroups of the symmetric groups show (see [6]). I n a particular group however, it m a y happen t h a t a m u c h stronger c o m m u t a t i o n law holds. Thus, if G e C F ( m , n, p) or G E N C F ( m ) (m > 3), a n d
[r~(G), rj(G)] < 7 f + , ~ ( a ) (i, j = 1, 2 . . . . ),
we say t h a t G has degree o/ commutativity k. (We do not assume t h a t k is the greatest such number).
The aim of the present p a r a g r a p h is to find conditions under which a group G has degree of c o m m u t a t i v i t y greater t h a n 0, t h a t is,
[r,(G), rj(G)] < 7,§247 (i, i = 1, 2, ...).
I n particular, this requires t h a t
[r~(G), 7,(G)] ~< 7~+~(G) (i = 1, 2 . . . m - 2). (9) F o r i = 1 this is always true b y Theorem 2.6, and for i = 2 it is always true on account of the definition of ~I(G). For 2 ~< i ~ < m - 2, (9) simply asserts t h a t ~l(G)/yl+2(G) is the eentraliser of ~(G)/~t+~(G) in G/~+~(G), for, in the case of the T-groups this centraliser cannot be larger t h a n a m a x i m a l subgroup, and in the case of the groups of NCF(m) this eentraliser m u s t have infinite index b y L e m m a 2.5. Our n e x t result shows t h a t whether
(9) holds or not is the crux of the matter.
T H e O R E m 2 . 8 . / ] G 6 CF (m, n, p) or G 6 N C F ( m ) ( m > 3 ) , a n d [~(G), ~(G)] ~<~+2(G) (i = 1, 2 . . . . , m - 2), then G has degree o / c o m m u t ~ i v i t y greater than O.
This is proved b y means of the following ]emma.
5 - 583801. Acta mr~hematica 100. Imprim~ le 26 septembre 1958.
58 N. BLACKBURN
LEMM.~ 2.9. Sutrpose that G E C F ( m , n, p) or O E N C F ( m ) (m > 4), and that
G/~/m_l (G)
has degree o/commutativity greater than O. Let elements s, s~ o / G be de/ined by the praperties that s and ~1 (G) generate G, s 1 and ~m-~ (G) generate ~1 (G). Write si =~t (s, sl) (i = 1, 2 . . .
m - 2). T h e n / o r i = 2, 3, ..., m - 2, st and ~1+1 (G) generate ~i(G), and
[81, 8m--2] = [82, 8m__3] -1 . . . [8t, 8m__i__l] (-1)1-1 . . . [mrn_2, 81] (-1)m-1.
N o t e t h a t there is no a m b i g u i t y in t h e definition of s I. B y T h e o r e m 1.1 y2(G) is generated b y Ya (G), Is1, s] = s2, a n d certain other c o m m u t a t o r s , one of whose c o m p o n e n t s lies in ~m_2(G), for G is generated b y s, s 1 a n d Sm_2(G). Since ~m_2(G) =~I(G), these other c o m m u t a t o r s already lie in ~a(G), a n d so for i = 2, st a n d ~t+l(G) generate yt(G). F o r i > 2 we prove this b y i n d u c t i o n on i: t h u s b y t h e inductive hypothesis st_~ a n d ~t (G) generate
~t_l(G). Hence b y T h e o r e m 1.1 yt(G) is generated b y yt+l(G), [St-l, s] =st, a n d certain other c o m m u t a t o r s , one of whose c o m p o n e n t s lies in Yl (G). B u t since G/~'m-1 (G) has degree of c o m m u t a t i v i t y greater t h a n 0 a n d i < m - 2,
[Yt--1 (G), Yl (O)] < ~t+1 (O),
a n d so these other c o m m u t a t o r s a l r e a d y lie in yl~l(G), a n d t h e result is proved.
F o r 2 ~<i ~ < n z - 2 ,
[8t, 8 m _ t _ l ] = 8 ~ 1 8 ~ m - | - 1 = 8~ 1 [8t_1, 8]srn-t 1 = 8~ 1 [8~mi-$-1 ' 8sm--I - 1].
Since G/ym_ 1 (G) has degree of c o m m u t a t i v i t y greater t h a n 0, [st-l,
8m--t--l]
E ~/m--1(G),
a n dso s~ml-t-1 is t h e p r o d u c t of St-x a n d an element of t h e centre of G. Also s 8m-t-x = SSml-t, a n d so
[8i, 8m--t--l] = 8~ 1 [81_1, 8 8 m l - t ] .
W o r k i n g o u t t h e r i g h t - h a n d side b y means of (1) a n d (2), a n d using t h e fact t h a t [St-x, Sm-t]
lies in t h e centre of G, we o b t a i n
[St, Sin-i-l] = [8t_1, 8re_t] -1, as required.
T h e o r e m 2.8 is trivial for m = 4, a n d we prove it for qn > 4 b y i n d u c t i o n on m. Apply- ing t h e inductive hypothesis to G/ym-I (G), we find t h a t O/ym-1 (G) has degree of c o m m u - t a t i v i t y greater t h a n 0. T h u s we have o n l y to prove t h a t [yi(G), ~m-t-l(G)] = 1 (i = 1, 2, .... m - 2). T h e conditions of L e m m a 2.9 are satisfied, a n d so in the n o t a t i o n there adopted,
[8t, 8rn-|-l] : [81, 8m_2] (-1)t-1.
B u t b y hypothesis, [~1 (G), ym_~(G)] = 1, a n d so st a n d sin-t-1 c o m m u t e . I t is clear from
O N A S P E C I A L C L A S S O F p - G R O U P S 59 T h e o r e m 1.2, Corollary 2 that sl c o m m u t e s with all sj for which j i> m - i, a n d that sm-i-1 c o m m u t e s with all sj for which j >~ i + I, and to [?~(G), ?m-i-1 (G)] = l, as required.
L e m m a 2.9 also has the following consequence.
T H ~ . O R E M 2.10. Suppose that G E C F ( m , n , p ) or G E N C F ( m ) (m>~5), and that G/?m-I (G) has degree o/commutativity greater than O. Then
(i) i / m is odd, G has degree o/commutativity greater than 0;
(ii) i/ m is even, G has degree o/commutativity greater than 0 i/ and only i / ? j m-1 (G) is Abelian.
B y T h e o r e m 2.8 G has degree of c o m m u t a t i v i t y greater t h a n 0 if a n d o n l y if [71 (G), ?m-s (G)] = 1, a n d in the n o t a t i o n of L e m m a 2.9 this is equivalent to [sl, sin_2] = 1. B y L e m m a 2.9 this is so if a n d only if Isis_l, s ~ ] = 1 when m is even, or [st(~_l), s89 = 1 when m is odd. This condition is always satisfied when m is odd, a n d when m is even it is satisfied if a n d only if [?tm_l(G), ?j~(G)] = 1, which is t h e condition for ?~m_l(G) t o be Abelian, according to L e m m a 2.1.
C o R O L L A R u I / G e CF (m, n, p) or G e N C F (m) (m ~> 5), and ?~ (G) is Abelian, then G has degree of commutativity greater than O.
This follows b y a v e r y simple i n d u c t i o n on m.
W e n o w reach t h e m a i n result of this p a r a g r a p h .
T H E O R E M 2.11. I / G E C F ( m , n , p ) , where m is odd and 5 ~ < m ~ < 2 p + l , or i/
G E N C F (m), where m is odd and m >1 5, then (7 has degree o/commuta~ivity greater than O.
l%r m = 5, this is a direct consequence of T h e o r e m 2.10 (i), for G/?4(G ) necessarily has degree of c o m m u t a t i v i t y greater t h a n 0. F o r m > 5 we proceed b y i n d u c t i o n on m.
A p p l y i n g t h e inductive hypothesis to G/?,~_s(G) we see t h a t this g r o u p has degree of c o m m u t a t i v i t y greater t h a n 0. Hence we m a y a p p l y L e m m a 2.9 to G/ym-1 (G), a n d de- fining s, sl, s 2 .. . . . s~_ a as in this l e m m a we find t h a t
[8i, 8j] e ~2f,+.~ , 1 ( a ) (i + j < m -- 3), (10)
[Si, am_t_2] ~ [81,
Sin_3]
(-1)$-1 (rood ?m_l(G)) (i = 1, 2 . . . m - 3). (11) N o w if G/?,,_ 1 (G) has degree of c o m m u t a t i v i t y greater t h a n 0, we can a p p l y T h e o r e m 2.10, a n d since m is odd, we o b t a i n a t once t h e required result. W e shall therefore assume t h a t G/?m_~ (G) does n o t h a v e degree of c o m m u t a t i v i t y greater t h a n 0 a n d o b t a i n a contra- diction. B y Theorem 2.8 it follows f r o m this a s s u m p t i o n t h a t (9) does n o t hold for i = m - 3, a n d t h u s t h a t ?l(G)/?,,_l(G) is n o t t h e centraliser of ?m_a(G)/?m_l(G). B u t ?l(G) is gene- r a t e d b y s 1 a n d $~_2(G): t h u s s I c a n n o t lie in this centraliser, since Sm_z(G) certainly does60 N. BLACKBURN
lie in it. A n d since 7m_3(G) is generated b y sin-3 and 7m_2(G) it follows t h a t [sm-s, sl] = Sin_9 say lies in 7m_2(G) b u t n o t in 7"_1(G).
B y (11) [S~,Sm_4]--Sm_ 2 (mod7~_1(G)),
and so [sin-2, Sx] ~ [s2, Sm-4] -1 Is2, Sm-4] ~'.
B u t [s 2, s"_4]" = [s~.', a~_,] = [sz[s~, sx], '"-4[s~-4, sl]]-
Also [s~, sl] lies in 74(G) and thus c o m m u t e s with a n y element of 7"_4(G): hence b y (1)
Similarly [sm_4, sl] ETm_z(G) and t h u s c o m m u t e s with sv Hence
[,~, ..,_41 s' = [.~, sm-~l,
a n d [sin-=, Sl] = 1. (12)
Since s x and ~m_~(G) generate 71(G), it follows t h a t sm-~ commutes with all elements of 7x(G). N o w s"_ 2 lies in 72(G) b u t not in 7m_l(G), and so b y T h e o r e m 2.4 s"_9, does n o t lie in ~, (G). H e n c e s~_~ cannot c o m m u t e with s, and the element sm-x = [s"-2, s] is not the unit element.
N e x t we prove b y induction on i t h a t
o ( _ , ) I - I ( , _ i ) (i = 2, 3 . . . . m - 3). (13)
[s,, sin-,-1] = ore- x F o r i = 2 we h a v e
[sa, S~_s] = s2-x S~-ss2 = s~l[sx, s] " ' - s = s ~ - l [ s ~ m - s , s ' m - ~ ] = s ~ l [ s , [ s x , s~_a],s[s,s~a_s]]. (14) N o w [s, s"_s] lies i n 7m_~(G) and thus c o m m u t e s with s 1 and all elements of 79.(G) b y (12).
B y (1) and (2) it follows from (14) t h a t
[s~, s~_3] = s~ x [sxs~l-~, s] = s;,l_x, as required. F o r i > 2 we h a v e b y the inductive hypothesis
__ ~ ( - 1)t(t--2) [ 8 | _ 1 , 8 r n _ t ] - - o " - 1 9
N o w
Isis S m _ L _ I ] -~_ S~ -x s~m-.t- X == S~ -1 [ 8 f _ l , S i s " - I - - 1 = S 7 1 fsSm--f-1 8s"-t-X] L t _ ,
^--I r^ ~ ( - I ) / - 1
= 8 ; x [81_ x [S~_x, 8"_~._X] , 8 [8, 8 " - I - I ] 1 = ~I t~i--I ~ " - 2 , S S m l - d
b y (11). Using (I) and (2) it follows t h a t
Is,, s"_,-d = [S,_x, s,._,]-' Is,._., s]~ -x~'-* -= s~-_"~ -x._l,, as required.
ON A SPECIAL CLASS OF p-GROUPS 61
Putting i = 89 - 1) in (13) we obtain s~(Yi -3) = 1, and from above sm_l~ = 1. This is impossible if GENCF(m). I f G E C F ( m , n, p), it implies t h a t 8 9 (rood p), and since m > 3, we have m > / 2 p § 3. This contradicts the hypothesis of the theorem and so the result is proved.
COROLLARY.
I/G
ECF(m, n, p), where m is even and 6 <. m <<. 2 p + 2, or i/ GENCF (m) where m is even and m >~ 6, then G has degree of commutativity greater than 0 i~ and only i/789 (G) is Abelian.
F o r we m a y apply Theorem 2.11 to G/7m_I(G), which shows t h a t this group has degree of c o m m u t a t i v i t y greater t h a n 0. The result therefore follows b y Theorem 2.10.
We mention the following consequences of Theorem 2.11.
T H E O R E M 2.12. Suppose that G is a p-group and that there exists an even integer m satisfying 4 <~ m ~ 2 p, such that Glum(G) ECF(m, n, p)/or some n. Then (~'m(G): ~',~ I(G)) ~ p . T H E O R :E M 2.13. Suppose that G is a nilpotent group, and that there exists an even integer m >~ 4, such that G/~'m (G)E NCF (m). I/T/~'m+l (G) is the torsion subgroup o/~,, ( G ) / ~ + 1 (G), then Ym ( G ) / T is cyclic.
L e t s, s I be defined in the usual way, and let x be an element defined b y the p r o p e r t y t h a t x and 7~(G) generate 7~_1(G). B y Theorem 1.1 and Theorem 1.2, Corollary 1, ~ ( G ) is generated b y ~ + 1 (G) and the elements [x, s], [x, Sl]. Let N be the subgroup of G genera- ted b y [x, s] and ym+l(G). Then y , , ( G ) / N is cyclic.
I f G is a p-group, suppose t h a t y,~(G)/N is of order p. Then G / N E C F ( m + 1, n + 1, p), and b y Theorem 2.11 G I N has degree of c o m m u t a t i v i t y greater t h a n 0. Thus Ix, sl] E/V, which is a contradiction. Hence N = 7~ (G), t h a t is, ym (G) is generated b y Ix, s] and 7~+1 (G).
The result now follows a t once from Theorem 1.5 (if).
Similarly, for Theorem 2.13, we obtain a contradiction if we assume t h a t ym(G)/2I is infinite~ and so ~m (G)/y,n+l (G) is an extension of a cyclic group b y a finite cyclic group.
Hence the result.
We shall not investigate further the conditions under which a general group of CF (m, n, p) has degree of c o m m u t a t i v i t y greater t h a n 0, b u t it will be proved later t h a t the groups of E C F (m, n, p) for which m > p + 1 have this property. We now wish to con- struct examples which show t h a t nothing more can be p r o v e d in this direction; t h a t is, we construct a p-group of m a x i m a l class of order p~r, where 6 <~ 2 r ~< p + 1, which does not have degree of c o m m u t a t i v i t y greater t h a n 0. Thus, for p > 3 and 3 ~< r ~< 89 § 1), let E be an elementary Abelian group of order pr, with generators el, e2,.., e~. L e t A bc the subgroup of the holomorph of E consisting of all elements which induce in E an auto- morphism of the form
62 ~r BLACKBURN
et-->e,e~t (i = 1, 2 . . . r - 1), er--~e, L e t ax, a 2 .. . . . at-1 b e t h e a u t o m o r p h i s m s in A d e f i n e d b y
ca,_ ~ ~(_l~r+t-1 (~_Jl 1) (i = 1, 2, r - 1; i = 1 . 2 , r - 1), e2J = er (?" = 1, 2 . . . r - 1).
N o w if b is a n y a u t o m o r p h i s m in A o t h e r t h a n 1 a n d e~ = e,e~ (i = 1, 2 . . . r - 1), define k t o be t h e g r e a t e s t of t h e i n t e g e r s 1, 2 . . . r - 1 such t h a t ak * 0 ( m o d p); t h e n ba(/_~ ~+%~' l e a v e s i n v a r i a n t all e~ for w h i c h i ~> k. I t follows t h a t t h e a u t o m o r p h i s m g r o u p A / E is g e n e r a t e d b y a~, a 2 .. . . . a,_ 1. W e h a v e
[a,, aj] = 1 ( l < i < j < r - 1 ) , a [ = l ( i = 1 , 2 . . . r - l ) . I t is e a s i l y verified t h a t A possesses a n a u t o m o r p h i s m ~ for w h i c h
a~ = a~ ai-11 ( i = 1 , 2 . . . r - 2 ) , ar- -- a~_l e~ 1, e~ = e i et-+ll (i = 1, 2 . . . r - 2), er--1 = er_ 1, er - er.
is of o r d e r p, a n d we m a y t h u s f o r m a n e x t e n s i o n G of A , such t h a t G / A is of o r d e r p, a n d a n e l e m e n t of G i n d u c e s t h e a u t o m o r p h i s m ~ in A (see [13], K a p . I I I , w 7). G is of o r d e r p2r a n d class 2 r - 1 , b u t does n o t h a v e d e g r e e of c o m m u t a t i v i t y g r e a t e r t h a n 0, since e l i = er-1.
T h e a b o v e c o n s i d e r a t i o n s o n l y m a k e use e s s e n t i a l l y of r e l a t i o n s b e t w e e n c o m m u t a t o r s a n d i t is t h e r e f o r e t o b e c o n j e c t u r e d t h a t t h e c a l c u l a t i o n s can be a p p l i e d m o r e g e n e r a l l y . I f for i n s t a n c e we consider n i l p o t e n t g r o u p s whose lower c e n t r a l factors a r e cyclic of a r b i t r a r y order, g e n e r a l i z a t i o n s of t h e a b o v e t h e o r e m s c a n b e o b t a i n e d . T h e r e s u l t s are, h o w e v e r , r a t h e r c o m p l i c a t e d ; t h e price of g e n e r a l i t y is a v e r y c o n s i d e r a b l e loss of c l a r i t y , a n d we h a v e t h e r e f o r e b e e n c o n t e n t t o s t a t e t h e r e s u l t s in t h e i r s i m p l e s t forms.
W e c o n c l u d e t h e p r e s e n t p a r a g r a p h b y o b t a i n i n g a r e s u l t o n t h e m a x i m a l n u m b e r of g e n e r a t o r s of t h e d e r i v e d g r o u p of a g r o u p of C F (m, n, p). W i t h l a t e r a i m s in m i n d i t will b e c o n v e n i e n t t o w o r k u n d e r s l i g h t l y m o r e g e n e r a l h y p o t h e s e s t h a n a r e n e c e s s a r y for t h i s p u r p o s e . T h u s we s u p p o s e t h a t G E C F ( m , n, p) (m > 3), a n d t h a t if m > 5, G/yz_I(G) has d e g r e e of c o m m u t a t i v i t y g r e a t e r t h a n 0. B y L e m m a 2.5 t h e c e n t r a l i s e r K of Yz-2 (G) in G is of i n d e x p a n d c o n t a i n s Sm_2(G). T h e r e a r e t h e r e f o r e a t l e a s t pm-,,(p _ 1)9. e l e m e n t s of G which b e l o n g n e i t h e r t o K n o r t o Yl (G): l e t s be such a n e l e m e n t , a n d l e t S be t h e cen- t r a l i s e r of s in G. C l e a r l y ~ - 1 (G) ~ S, a n d we p r o v e t h a t S f3 Y2 (G) = ~'z-1 (G). I f in f a c t
O N A S P E C I A L C L A S S O F p - G R O U P S 63 there exists an element x of S N ~ (G) which does not belong to Ym-1 (G), then there exists an integer i, with 2 ~<i ~ m - 2 , such t h a t x lies in ~t(G) but not in ~t+l(G). Thus x a n d
~+1 (G) generate ~t (G); but s was chosen so t h a t it does not belong to the centraliser of
~t(G) modulo yt+2(G), and so [x, s] cannot lie in ~ . ~ (G). Since i ~< m - 2, and x E S , this is a contradiction.
L E M ~ A 2.14. Suppose that G E C F ( m , n, p) (m > 3) and that, i / m > 5, G//~,~_I(G) has degree o/commutativity greater than O. Let s be an element of G which belongs neither to ~1 (G) nor to the centraliser o/7m-2 (G), and let S be the centraliser o / s in G. Then S N ~2 (G) = 7m_t(G).
N e x t we show t h a t we can proceed from 72 (G) to ~m-2 (G) b y adjoining elements of S.
T ~ E O R E M 2.15. Under the conditions o/ Lemma 2.14, a set T o~ elements o / S can be chosen, which together with y2(G) generate Sm_2(G). 11 t, u E T, then [t, u] eTm_l (G).
This m a y be proved b y induction on m. Thus for m > 4, there exists a set T of ele- ments which together with 72(G) generate ~ - 2 (G), such t h a t if t E T, then [s, t] ETa_ I (G), as is seen b y applying the inductive hypothesis to G/Tm_ 1 (G). This is also true for m = 4, since Sm_2(G) =~{G). Let x be an element which together with 7~_1(G) generates 7~_2(G), so t h a t b y the definition of s, y = [x, s] * 1, t h a t is, y generates 7~_t(G). Thus for each element l E T , there exists an integer a such t h a t [s, t] = y~; we p u t t = ~x ~, and denote b y T the set of all elements t which arise in this way. Since m ~> 4, t ESm_2(G), and so ~m_2(G) is generated b y T and 72(G). Since also b y (1) and Theorem 1.4
[s, t] = [s, e x ~] = [8, x ~] y ~ = 1,
we see t h a t T is contained in S. Finally, if t, u E T, t h e n [t, u] ES N ~,2(G), a n d so It, u] e
~ _ I ( G ) , b y L e m m a 2.14.
We shall need one or two consequences of Theorem 2.15.
C o R o L ~. A R u 1. Under the same conditions S is o/order p~- "~ "2 and is o/class at most 2.
First of all we show t h a t S N ~_~ (G) is generated b y T and ~ - I ( G ) , and is of order
~0 ~-m+l. I f x ~ S fi ~ _ ~ ( G ) , z can be written in the form yz, where y is a product of elements of T, a n d z ~ ( G ) , for ~_~.(G) is generated b y T a n d ~ ( G ) . Since T is contained in S, it follows t h a t y E S: thus z = y - t x ~S. Hence z ~ - 1 (G) b y L e m m a 2.14. Also, if t e T a n d t is of order p~ modulo ~ ( G ) , then t ~ e ~ ( G ) N S =~m_l(G). Thus, since ~m_~(G)/~(G)is of order p~-~n, it follows t h a t T a n d ~,m_t (G) generate a group of order ~ - ~ + ~ , for [T, T] ~<
~ _ ~ (G). This proves the assertions.
N e x t we show t h a t S (I ~t(G) = S N ~_2(G). I f x E S N ~I(G) and s t is an element of G so defined t h a t s t a n d ~m-~ (G) generate ~t (G), then we can write x - s ~ y where y ~ _ ~ (G) for a suitable integer a. Thus
6 4 ~r BLACKBURN
1 = [s, x] = [s, s ~ y ] = [s, y] [s, s~] ~.
B u t
[8, y]Eya(G),
since ~m_2(G)=~(G), and so [s,s~]E?a(G ). Hence, modulo Ta(G) sr commutes with every element of G; t h a t is, sTE~/(G)=~_2(G). Thus x E~_~(G), as required.Finally, if x ES, w e can write x = s~y where y E~I (G). Since x and s lie in S, we h a v e y E S N y l ( G ) = S N ta_2(G). Thus S is generated b y s, T and ?z-1(G). Also, since sVES N
?I(G) = S N $~_2(G), S is of order p"-~+~. S is of class 2, since IT, T] <<.y,,_l(G).
C 0 R 0 L L A R Y 2. Under the same conditions the conjngacy class o / G containing s is the coset sy2 (G).
F o r b y Corollary 1 s has p a - 2 distinct conjugates in G. Since s ~ = s i s , x] for a n y ele- m e n t x of G, each of these conjugates is of the form s y where y E~2(G ). And since ~2(G) has just p~-2 elements, it follows t h a t each element of this form m u s t be a conjugate of s.
THEOR~.M 2.16. Suppose that G ECF(m, n, p) (m > 3), and that r is the smallest positive integer such that there exists an element s o/ G, not belonging to ~I (G), /or which s vr lies in
~2 (G). Then T2 (G) can be generated b y / e w e r than p~ elements.
Suppose t h a t this is not true. Then the index of the Frattini subgroup ~b(y~(G)) of
~2(G) in ~2(G) is at least p rr, and so there exists a normal subgroup N of G, such t h a t
~(~2(G)) ~< N <~2(G), and ~ 2 ( G ) / N is elementary Abelian of order ppr. B y considering G//N we see t h a t without loss of generality, it m a y be assumed t h a t T2 (G) is an elementary Abelian group of order p f , for b y L e m m a 2.3 G / N E C F ( p ~ + 2, n', p) for some n ' . B y Theorem 2.10, Corollary, G has degree of c o m m u t a t i v i t y greater t h a n 0. Hence L e m m a 2.14 m a y be applied, a n d since s vr E S N ~2 (G), it follows t h a t s vr E ym-1 (G).
I f s I is an element which together with ~m-2 (G) generates $1 (G), we define st = a~ (s, sl) (i = 2, 3 . . . . , m - 1), so t h a t as in L e m m a 2.9, si a n d ~t+l(G) generate ~t(G) (i = 2 , 3, . . . . m - 1). We prove b y induction on k t h a t
. .
81 ~ tk . . . 8 k + 1 "~
F o r k = 1 this is clear; for k > 1 we assume the corresponding result for k - 1 a n d trans- form b y s, using s~ =sts~+ 1. Since ?2(G) is Abelian, we m a y deduce the stated result at once. Putting k = p ' = m - 2, and using the fact t h a t ~,2(G) is of exponent p, we obtain
8~ pr ~ 8 1 8 r a _ l .
Thus s vr cannot belong to the centre of G. B u t we have shown t h a t s vr E ~ _ I ( G ) and so we have a contradiction. Thus Theorem 2.16 is proved.
ON A SPECIAL CLASS OF p-FROUPS 6 5
I t is v e r y easy to show t h a t Theorem 2.16 is the best possible result. Thus let E be an elementary Abelian group of order p~' generated b y elements sl, s 2 .. . . . sr~. L e t a be the automorphism of E defined b y s~ = s~st+z (i = 1, 2 . . . pr _ 1) a n d s ~ -- spr. Then a is of order p ' and we m a y form an extension G of E, such t h a t G / E is cyclic of order p ' and an element s of G induces the automorphism a in E. Then G E C F ( p r + 1, 2r, p), and
~, (G) cannot be generated b y fewer t h a n p ' - 1 elements. I t is to be observed t h a t in the case r = 1 the group t h a t we have constructed is the Sylow p-subgroup of the symmetric group of degree p~.
We mention two particular eases of Theorem 2.16.
COROLLARY 1. I / GECF(m, n, 2), and there exists an element s o / G which does not belong to ~I (G), such that s2E~2(G), then ~2(G) is cyclic.
COROLLARY 2. I / G E C F ( m , n, 3), and there exists an element s o / G which does not belong to ~1 ( G), such that s a E~2 ( G), then ~ ( G) is an A belian group ugth at most two generators.
The first corollary is obvious. To prove the second we observe first t h a t ~2(G) can be generated b y 2 elements, b y Theorem 2.16. I t follows t h a t [~2(G), ~3(G)] = 1 (see [1], Theorem 2). Since ~2 (G)/~a (G) is cyclic, it follows from L e m m a 2.1 t h a t ~2 (G) is Abelian.
3. The corollaries of Theorem 2.16 suggest t h a t far deeper results will be obtainable if some condition is imposed on G / ~ ( G ) , and we shall therefore assume henceforth t h a t this group is elementary Abelian, t h a t is, t h a t G E E C F ( m , n, p). For m > 3 such a group always possesses a subgroup of order pm a n d class m - 1 with the same lower central series as G. F o r b y the results of the previous p a r a g r a p h G can be generated by two ele- ments x, y and ~ (G). Thus, if H is the group generated b y x, y, H ~ (G) = G, and so b y Theorem 1.3, ~ ( H ) = ~ t ( G ) . B y hypothesis, x ~, y~ are elements of ~ 2 ( G ) = ~ 2 ( H ) , and so H is of order pro, as required. Thus the s t u d y of the groups of E C F ( m , n, p) reduces essentially to t h a t of p-groups of m a x i m a l class, a t least so far as the properties of the lower central series are concerned. The following l e m m a shows t h a t the groups of E C F (m, n, p) possess other subgroups which are p-groups of maximal class.
LV.MMA 3.1. Suppose that G E E C F ( m , n, p) (m > 3) and that, when m > 5, G/~m_I(G) has degree o/ commntativity greater than O. Then G possesses a subgroup K which is of order pm-~and clazs m - 2 , and ~l(K) = ~ t + l ( G ) (i = 1, 2 . . . m - 2 ) .
One of our principal aims is to prove t h a t if G E E C F (m, n, p) a n d m > p + 1, then G has degree of c o m m u t a t i v i t y greater t h a n 0 (Theorem 3.8). This will be proved b y induc- tion on m and so, in L e m m a 3.1 a n d a n u m b e r of the following results, we shall work under the hypothesis t h a t G/~rn_ 1 (G) has degree of c o m m u t a t i v i t y greater t h a n O. This hypo-
6 6 N. BLACKBURN
thesis is of course shown t o be unnecessary b y Theorems 2.11 a n d 3.8. I n such a group our n o t a t i o n will be as follows, s denotes a n element which belongs neither to 71(G) n o r t o t h e centraliser of 7~_~(G). s~ denotes a n element which belongs t o 71(G) b u t n o t t o
~m_3(G). F o r i = 2 , 3 . . . . , m - 1 we write s~=a|(s, sl). B y L e m m a 2.9 st a n d 7t+l(G) generate 7t (G) if 2 < i ~< m - 2. This is also true if i = m - 1. F o r sin_ 2 a n d 7m-1 (G) generate
~'m-3 (G), a n d s c o m m u t e s with every element of 7m-1 (G) b u t does n o t belong to t h e een- traliser of 7m-3 (G) : t h u s s a n d Sin-2 do n o t c o m m u t e , t h a t is, s~_ 1 =~ 1. I t is also i m p o r t a n t t o observe t h a t s ~ ETm_I(G ). F o r since G/72(G ) is e l e m e n t a r y Abelian, s ' E 72(G); also s ~ certainly belongs t o t h e eentraliser of s, a n d so b y L e m m a 2.14, s ~ E 7~-1 (G).
To p r o v e L e m m a 3.1 we define K to be t h e group g e n e r a t e d b y s a n d 72(G). Since 72(G) is of order pro-2 a n d s ~ E 7m_l(G) ~< 73(G), K is of order pro-1. Also s 2 a n d s are b o t h elements of K a n d am_3(s, s~) = sin_ 1 r 1 ; t h u s K has class at least m - 2. Since m - 2 is t h e m a x i m a l class of a group of order pro-l, K is a p - g r o u p of m a x i m a l class a n d 7~(K) is of order p,,-~-i (i = 1, 2 . . . m - 2). Since K ~>73(G), K is a n o r m a l subgroup of G, a n d so 7~(K) is n o r m a l in G. F o r i / > 2 7~(K) -~<72(G), a n d so b y L e m m a 2.3, 7~(K) =Tt+l(G).
Also
[73 (G), 72 ( g ) ] = [73 (G), 73 (G)] ~< 75 (G) = 74 ( g ) ,
a n d so 7 2 ( G ) ~ 71(K). B u t these groups have t h e same order, a n d are therefore equal:
t h u s t h e result is proved.
W e observe t h a t a n y such subgroup K has degree of c o m m u t a t i v i t y greater t h a n 0, for
[7,(K}, 7j(K)] = [7,+1(G), 7j+~(G}] ~< 7,~j§ = r~+~+1 (K).
I t is also t o be observed t h a t b y repeated application of L e m m a 3.1, if 2 ~< r < m - 3, we can c o n s t r u c t a subgroup L of G of order pm-r+l a n d class m -- r, with degree of com- m u t a t i v i t y r - 1 , such t h a t 7 j ( L ) = 7 r + j _ I ( G ) ( j = 1, 2 . . . m - r ) . This result is clue to W i m a n ([12], w 4).
Our n e x t aim is to investigate t h e " p o w e r - s t r u c t u r e " of t h e g r o u p s of E C F ( m , n, p).
F o r small values of m, we o b t a i n t h e following result.
T H E O R E M 3.2. SupTose that G E E C F (m, n, p), where p is odd and 4 ~ m <. p + 1.
Then G/Tm_I(G ) and 72(G) are o/ exponent p. I] m <~ p, the elements o/ G o/ order at most p ]orm a characteristic subgroup o~ index at most p.
B y T h e o r e m 2.11 G/Tm_I(G ) has degree of c o m m u t a t i v i t y greater t h a n 0, a n d so there are at m o s t 2 m a x i m a l subgroups of G which can be centralisers of t h e groups 7j(G)/7~+2(G) (i = 2 , 3 . . . m - 2 ) . G has p + 1 m a x i m a l subgroups containing Sm_~(G), a n d so, since p is odd, we m a y select t w o such subgroups, neither of which is t h e een-
